consider f being PartFunc of D, the carrier of Y such that
A3: X = dom f and
A4: for p being Real st p > 0 holds
ex k being Nat st
for n being Nat
for x being Element of D st n >= k & x in X holds
||.(((H . n) /. x) - (f /. x)).|| < p by A1;
take f = f; :: thesis: ( X = dom f & ( for x being Element of D st x in X holds
for p being Real st p > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p ) )
thus X = dom f by A3; :: thesis: for x being Element of D st x in X holds
for p being Real st p > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p
let x be Element of D; :: thesis: ( x in X implies for p being Real st p > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p )
assume A5: x in X ; :: thesis: for p being Real st p > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p
let p be Real; :: thesis: ( p > 0 implies ex k being Nat st
for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p )
assume p > 0 ; :: thesis: ex k being Nat st
for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p
then consider k being Nat such that
A6: for n being Nat
for x being Element of D st n >= k & x in X holds
||.(((H . n) /. x) - (f /. x)).|| < p by A4;
take k = k; :: thesis: for n being Nat st n >= k holds
||.(((H . n) /. x) - (f /. x)).|| < p
let n be Nat; :: thesis: ( n >= k implies ||.(((H . n) /. x) - (f /. x)).|| < p )
assume n >= k ; :: thesis: ||.(((H . n) /. x) - (f /. x)).|| < p
hence ||.(((H . n) /. x) - (f /. x)).|| < p by A5, A6; :: thesis: verum